EQUATIONS OF MATHEMATICAL DIFFRACTION THEORY © 2005 by CRC Press LLC Differential and Integral Equations and Their Applications A series edited by: A.D. Polyanin Institute for Problems in Mechanics, Moscow, Russia Volume 1 Handbook of First Order Partial Differential Equations A.D. Polyanin, V.F. Zaitsev and A. Moussiaux Volume 2 Group-Theoretic Methods in Mechanics and Applied Mathematics D.M. Klimov and V. Ph. Zhuravlev Volume 3 Quantization Methods in the Theory of Differential Equations V.E. Nazaikinskii, B.-W. Schulze and B. Yu. Sternin Volume 4 Hypersingular Integral Equations and Their Applications I.K. Lifanov, L.N. Poltavskii and G.M. Vainikko Volume 5 Equations of Mathematical Diffraction Theory Mezhlum A. Sumbatyan and Antonio Scalia © 2005 by CRC Press LLC EQUATIONS OF MATHEMATICAL DIFFRACTION THEORY Mezhlum A. Sumbatyan Antonio Scalia CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2005 by CRC Press LLC TF1668 disclaimer.fm Page 1 Thursday, July 15, 2004 2:40 PM Library of Congress Cataloging-in-Publication Data Sumbatyan, Mezhlum A. Equations of mathematical diffraction theory / Mezhlum A. Sumbatyan, Antonio Scalia. p. cm. Includes bibliographical references and index. ISBN 0-415-30849-6 (alk. paper) 1. Diffraction—Mathematics. I. Scalia, A. II. Title. QC415.S95 2004 535'.42'0151—dc22 2004051957 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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Visit the CRC Press Web site at www.crcpress.com © 2005 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-415-30849-6 Library of Congress Card Number 2004051957 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper © 2005 by CRC Press LLC PREFACE The connection between heuristic and strictly formal methods is seemingly one of the most interesting and debatable questions in modern mathematics. Each of these two differentapproaches,whosefoundationswerelaidbySocratesandAristotle,respectively, andinthenewhistoryarere(cid:3)ectedindiscussionsandwrittenpapersbyDescartes,Leibnitz and Bacon, has its own intrinsic merits and restrictions. Moreover, a large number of discoveriesin science were made owing to a combination of strict and heuristic methods ofinvestigation. Apparently,oneofthebrightestexamplesinmodernmathematicalphysicsisdiffraction theory, where the combination of the two approaches would lead so ef(cid:2)ciently to such impressive results. Many important and interesting solutions and even some classical theories appeared from heuristic ideas, and the most impressive example was given by Kirchhoff’s physical diffraction theory, which is based upon a clear (cid:147)light and shadow(cid:148) concept for diffracted wave (cid:2)elds. Subsequently, many of these heuristic results were rigorouslysubstantiatedandprovedastheorems. Ontheotherhand,unsuccessfulattempts to prove some other heuristic ideas caused signi(cid:2)cant progress in the development of formal methods that yielded correct solutions, different sometimes from those prompted bysomeone’sintuition. The above speci(cid:2)c features have affected the style of presentation of the book. Each section deals with a discussion of heuristic ideas, which as a rule are substantiated (or disproved)with the use of rigorous mathematical methods. Due to limited volume of the book,atsomeplaceswegiveonlyabriefsketchofthesubstantiation,referringthereader totheoriginalliteratureformoredetails. Onemorespeci(cid:2)cfeatureofthepresentedmaterialisconnectedwiththerapidprogress in computertechnologyover the last 20 years, whichhas signi(cid:2)cantlychangedour view› point on what could be accepted as ef(cid:2)cient methods of investigation. Only recently, expansionofunknownfunctionsintoseriesintermsofspecialfunctions,whenaproblem reduced to in(cid:2)nite system of linear algebraic equationswith respect to coef(cid:2)cientsof the expansion, was regarded as a standard method. Such a (cid:147)semi›analytical(cid:148) approach was ef(cid:2)cient15(cid:150)20yearsago,whentheevaluationofregularityoftheobtainedin(cid:2)nitesystems seemedtobeveryimportant,sincethiscouldguaranteeaccuracyofasolutionbyretaining onlyfew(cid:2)rstequations,whichwasacceptablefor(cid:2)rst›generationscomputers. Nowadays, whenthereisnotmuchdifferencebetween10·10and500·500systemsevenfor home personalcomputers,suchaviewpointlooksarchaic,sincethetimerequiredtoconvertthe system to a form appropriate for (cid:147)fast computations(cid:148) is much greater than that for (cid:147)slow computations(cid:148)basedonmoderndirect numericalmethodslike boundaryelementmethod and (cid:2)nite element method. Apparently, it should be agreed that in the caseswhere direct numerical techniques provide reliable results in an acceptable computational time, they should be regarded as most ef(cid:2)cient for the problem in question. It is very important to recognizethe caseswhere one has a priori to reject direct numerical methods. Theseare listedbelow. (cid:14) 1 . Problems where an exact analytical solution or a good approximation to it can be obtained. Diffractiontheoryshowsmanyexamplesofthiskind. (cid:14) 2 . Studying dynamic processes with high frequencies. Here, one has to take at least © 2005 by CRC Press LLC Pagev 10 nodes per wavelength to obtain reliable results by any direct numerical method. As the wavelength decreases (i.e., the frequency increases) within a given frequency range, the total number of nodes increases very rapidly, which results in too large algebraic systems. An impressive example is given by room acoustics. Suppose a sound wave of frequencyf = 2kHz,whosewavelengthin air is 17cm,propagatesin a 17›m long room. For reasonable numerical accuracy, one should hence take at least 1000 nodes along the room length. If the room has a width of 8m and a height of 5.1m, one has to consider 1000·500·300»108 (cid:2)nite›elementnodesandperformcomplex›valuedarithmetic. This cannot be implemented even on the most powerful super computers. Here, a reasonable criterionforacceptabilityofanumericalapproachisitsimplementabilityonaPCorsimilar computer. So, obtaining solutions to such high›frequency problems in exact formulation bydirectnumericaltechniquesdoesnotseemtobefeasibleinthevisiblefuture. (cid:14) 3 . Studying phenomena of complex qualitative nature. Since direct numerical methods provideonlynumbers,whichare usuallytabulatedandplotted, it is often verydif(cid:2)cult to extract such complex qualitative effects from numerous tables and graphs. Instead, it is preferable to construct an approximate analytical solution, from which qualitative effects maybeextractedexplicitly. (cid:14) 4 . Cases where an exact analytical solution has been obtained but its representation is inapplicabletopracticefor speci(cid:2)ccalculations. Anexampleofthiskindisconsideredin Section6.1. In suchinteresting cases,oneshouldlookfor analternativeapproach,which isoftentheconstructionofanapproximatesolutionthatwouldbemoreappropriateforfast computationsthantheexactanalyticalsolutionobtained. The above situations are not widespread but, when met, are very dif(cid:2)cult to cope withef(cid:2)ciently,especiallyiftheresearcherdoesnothavesuf(cid:2)cientexperienceintackling them. Thispromptedustoconcludeeachsectionwithaspecialsubsectiontitled(cid:147)Helpful Remarks,(cid:148)whichmayhelpthereadertobuilduphisorherownlessformalconceptionand allowthecreationofamorecompletepictureoftheissueunderconsideration. Notethattheapplicationofnumericalmethodsinregularcasesiswelldescribedinthe classicalliterature. Forthisreason,weonlyconsidernumericalmethodsforsomeirregular operatorproblems;seeChapter9. To summarize, the main purpose of the present book is to show the close connection between heuristic and rigorous methods in mathematical diffraction theory. We focus on differentialandintegralequationsthatcaneasilybeutilizedinpracticalapplications. Such an approach is accounted for by the choice of our potential readers. The book presentsclearandelegantmethodsandisaimedatgraduateandpost›graduatestudents,so that they could quickly examine the state of the art in a speci(cid:2)c (cid:2)eld of interest. At the same time, researcherswith considerableexpertise in dealing with diffraction theory will hopefully discoverthat the time of clear explicit solutions in unsolvedcomplexproblems has not passed yet(cid:151)this is demonstrated by the authors’ original results in Sections 4.5, 4.6, 5.4(cid:150)5.7, and 6.3(cid:150)6.6 as well as in many sections of Chapters 7(cid:150)9. Furthermore, we hope that an experienced reader will be able to discover for him› or herself new helpful methods,bothanalyticalandnumerical. The reader will seein what follows that we prefer to rely upon classicalresults of the foundersof modernscienceunlikearatherwidespread(mistaken)pointof viewthatonly very complicated recent (cid:147)abstract(cid:148) theories can provide further progressin contemporary science. Westronglyrecommendtheyoungerreadertooperatewithclassicalmathematical theories, and the present book will demonstrate that the fruitful ideas of Hilbert, Cauchy, Fourier,Abel,Poisson,Weyl,Riemann,Green,Kirchhoff,Rayleigh,Helmholtz,Neumann, andotherscanguidethereaderveryef(cid:2)cientlyaroundpresent›dayproblemsindiffraction theory. It should also be stressedthat we tried to avoid too formal presentation, since we © 2005 by CRC Press LLC Pagevi believethat wielding thoroughknowledgein anymathematical theoryimplies applyingit effectivelyandsuccessfullyto practicerather than operatingwith the formal apparatusof thetheory. Duetoitslimitedvolume,anymonographcannotcoverallimportantquestions,andthe presentbookisnoexception. Forexample,thereaderwillnot(cid:2)ndheretransientproblems at all. The presentation is con(cid:2)ned to boundary problems for elliptic operators only, and only those with constant coef(cid:2)cients (except Section 3.6). Moreover, the main focus is on methods that provide solutions without too cumbersome mathematical manipulations. For example,the readerwill not (cid:2)ndthe structure of the wave(cid:2)eldin the (cid:147)semi›shadow(cid:148) zonein diffraction by convexobstacles,andin the method of (cid:147)edgewaves(cid:148)in diffraction fromlinearsegments,thereaderwillonly(cid:2)ndtheleadinghigh›frequencyasymptoticterm, whichisconstructedbyasimpleandeleganttechnique. The sections, displayed formulas, and (cid:2)gures are enumerated independently within eachchapterwiththechapternumberinfront. Thebookisintendedforthereaderfamiliarwithfundamentalsofreal,complex›valued, andfunctionalanalysiswithin a standardcourseoncalculusin the(cid:2)rst three yearsof any universityprogramofmathematical,physical,orengineeringdepartments. Thestyleandcontentofthisbookhavebeenin(cid:3)uencedbytheauthors’friends,teachers, and colleagues, Alexander Vatulyan (Rostov State University, Russia), Mauro Fabrizio (UniversityofBologna,Italy),andDorinIesan(UniversityofIasi,Romania). The authors are grateful to Alexander Manzhirov and Alexei Zhurov for their helpful discussionsandcomments. The(cid:2)rstauthoristhankfultohiswife,AngelaSumbatyan,andtohisdaughters,Laura, Carina,andAngelica,whoassistedandinspiredhiminthewritingofthebook. M.A.Sumbatyan A.Scalia © 2005 by CRC Press LLC Pagevii AUTHORS MezhlumA.Sumbatyan,Ph.D.,D.Sc.,isanotedscientistinthe(cid:2)eldofwavedynam› ics,diffractiontheory,andill›posedproblems. MezhlumSumbatyangraduatedfromtheFacultyofMechanicsandMathematics,Ros› tovStateUniversity,Russia,in1969andreceivedhisPh.D.degreein1980attheInstitute forProblemsinMechanics,RussianAcademyofSciences,Moscow. HisPh.D.thesiswas devotedtoasymptoticmethodsfor solvingintegralequationsarisingin someproblemsof mechanicsandacousticswith mixedboundaryconditions. In 1995,ProfessorSumbatyan receivedhisDoctorofSciencesdegree;hisD.Sc.thesiswasdedicatedtodirectandinverse problemsofdiffractiontheory,withapplicationstoreconstructionofobstaclesbyultrasonic techniques. In1985(cid:150)2000,MezhlumSumbatyanworkedintheResearchInstituteofMechanicsand Applied Mathematics of the Rostov State University. Since 2001, Professor Sumbatyan hasbeenamemberofthestaffoftheFacultyofMechanicsandMathematics,RostovState University. Professor Sumbatyan has made important contributions to new methods in the theory andanalyticalmethodsappliedtodirectandinversediffractionproblems. Heisanauthor ofmorethan120scienti(cid:2)cpublications,amemberoftheRussianAcousticalSociety,and amemberoftheAcousticalSocietyofAmerica. Address: FacultyofMechanicsandMathematics E›mail: [email protected] ZorgeStreet5 344090Rostov›on›Don,Russia Antonio Scalia, Ph.D., D.Sc., is a prominent scientist in the (cid:2)eld of mathematical physics. Antonio Scalia graduated from the Faculty of Mathematics, Physics and Natural Sci› ences, University of Catania, Italy, in 1972 and received his Ph.D. degree in 1980 at the UniversityofCatania. Histhesiswasdevotedtosolvabilityanduniquenessinmathematical formulationofmechanicalandacousticalproblemsformediawithmicro›structure. Since 1989, Professor Scalia has been a member of the staff of the Department of Mathematics and Informatics at the University of Catania as an Associate Professor, and since2001,asaFullProfessor. ProfessorScaliahasgreatexpertisein thetheoryof viscoelasticity,electromagnetism, and acoustics. He has signi(cid:2)cant achievements in the mixture theory in solids, with applications to wave processesin elastic and acoustic media. Recently, within the theory ofporousmedia,hedevisedanewapproachtotheanalysisofwavepropagationinelastic solids with pores. He explicitly studieda characteristic equationof the porouscomposite mediaappliedtothelayeredgeometry. ProfessorScaliaisanauthorofnearly100scienti(cid:2)c publications. Address: DipartimentodiMatematicaeInformatica E›mail: [email protected] Universita· diCatania,VialeA.Dorian.6 95125Catania,Italy © 2005 by CRC Press LLC Pageix CONTENTS Preface Authors Contents 1. SomePreliminariesfromAnalysisandtheTheoryofWaveProcesses 1.1. FourierTransform,LineIntegralsofComplex›ValuedIntegrands,andSeriesin Residues 1.2. ConvolutionIntegralEquationsandtheWiener(cid:150)HopfMethod 1.3. SummationofDivergentSeriesandIntegrals 1.4. AsymptoticEstimatesofIntegrals 1.5. FredholmTheoryforIntegralEquationsoftheSecondKind 1.6. FredholmIntegralEquationsoftheFirstKind 1.7. SingularIntegralEquationswithaCauchy›TypeSingularityintheKernel 1.8. Hyper›SingularIntegralsandIntegralEquations 1.9. Governing Equations of Hydroaeroacoustics, Electromagnetic Theory, and DynamicElasticity 2. Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium 2.1. PropertiesofthePotentialsofSingleandDoubleLayers 2.2. BasicIntegralEquationsoftheDiffractionTheory 2.3. Propertiesof IntegralOperatorsof Diffraction Theory: GeneralCaseandLow Frequencies 2.4. FullLow›FrequencySolutionforSphericalObstacle 2.5. Application: ScatteringDiagramforObstaclesofCanonicalShape 2.6. AsymptoticCharacteroftheKirchhoffPhysicalDiffractionTheory 3. WaveFieldsinaLayerofConstantThickness 3.1. Wave Operator in Acoustic Layer: Mode Expansion, Homogeneous and InhomogeneousWaves 3.2. PrinciplesofSelectionofUniqueSolutioninUnboundedDomain 3.3. WavesinElasticLayer 3.4. Generalized Riemann’s Zeta Function and Summation of Some Oscillating Series 3.5. Application: Ef(cid:2)cient Calculation of Wave Fields in a Layer of Constant Thickness 3.6. WavesintheStrati(cid:2)edHalf›Plane © 2005 by CRC Press LLC Pagexi 4. AnalyticalMethodsforSimplyConnectedBoundedDomains 4.1. GeneralSpectralPropertiesoftheInteriorProblemforLaplacian 4.2. ExplicitFormulasforEigenfrequenciesofRoundDisc 4.3. SomeVariationalPrinciplesforEigenvalues 4.4. Weyl(cid:150)CarlemanTheoryofAsymptoticDistributionofLargeEigenvalues 4.5. ExactExplicitResultsforSomePolygons 4.6. ExplicitAnalyticalResultsforSomePolyhedra 5. IntegralEquationsinDiffractionbyLinearObstacles 5.1. IntegralOperatorsinDiffractionbyLinearScreenandbyaGapintheScreen 5.2. Operator Equation in Diffraction Problem on a Crack in Unbounded Elastic Medium 5.3. High›FrequencyAsymptoticsinDiffractionbyLinearObstaclesinUnbounded Medium 5.4. High›Frequency Asymptotics for Diffraction by Linear Obstacles in Open Waveguides 5.5. High›FrequencyDiffractionbyaLinearDiscontinuityintheWaveguide 5.6. WavesinElasticHalf›Space. FactorizationoftheRayleighFunction 5.7. IntegralEquationoftheMixedBoundaryValueProblemforElasticLayer 6. Short›WaveAsymptoticMethodsontheBasisofMultipleIntegrals 6.1. Schoch’sMethod: ExactRepresentationof3DWaveFieldsbyOne›Dimensional Quadratures 6.2. High›FrequencyWaveFieldsinElasticHalf›Space 6.3. AsymptoticNatureoftheGeometricalDiffractionTheory 6.4. High›FrequencyDiffractionwithRe›Re(cid:3)ections 6.5. Application: ExamplesofHigh›FrequencyMultipleDiffraction 6.6. Application: PhysicalDiffractionTheoryforNonconvexObstacles 6.7. Short›WaveIntegralOperatorinDiffractionbyaFlawinElasticMedium 6.8. High›Frequency Asymptotics of Integral Operator in a Three›Dimensional DiffractionTheory 7. InverseProblemsoftheShort›WaveDiffraction 7.1. Some Basic Results in a Local Differential Geometry of Smooth Convex Surfaces 7.2. ReducingInverseProblemoftheShort›WaveDiffractiontoMinkowskiProblem 7.3. ExplicitResultsforaDifferentialOperatorofthe2DInverseProblem 7.4. ExactExplicitInversionoftheBasicOperatorintheCaseofAxialSymmetry 7.5. NonlinearDifferentialOperatoroftheThree›DimensionalInverseProblem 7.6. Reconstruction of Nonconvex Obstacles in the High›Frequency Range: 2D Case 7.7. Reconstruction of Nonconvex Obstacles in the High›Frequency Range: 3D Case © 2005 by CRC Press LLC Pagexii